Locally compact vector spaces and algebras over discrete fields

Abstract

The structure of locally compact vector spaces over complete division rings topologized by a proper absolute value (or, more generally, over complete division rings of type V) is summarized in the now classical theorem that such spaces are necessarily finite-dimensional and are topologically isomorphic to the cartesian product of a finite number of copies of the scalar division ring [9, Theorems 5 and 7], [6, pp. 27-31]. Here we shall continue a study of locally compact vector spaces and algebras over infinite discrete fields that was initiated in [16]. Since any topological vector space over a topological division ring K remains a topological vector space if K is retopologized with the discrete topology, some restriction needs to be imposed in our investigation, and the most natural restriction is straightness: A topological vector space E over a topological division ring K is straight if for every nonzero a E E, fa: A -Aa is a homeomorphism from K onto the one-dimensional subspace generated by a. Thus if K is discrete, a straight K-vector space is one all of whose one-dimensional subspaces are discrete. Any Hausdorff vector space over a division ring topologized by a proper absolute value is straight [6, Proposition 2, p. 25]. In ?1 we shall prove that if E is an indiscrete straight locally compact vector space over an infinite discrete division ring K, then K is an absolutely algebraic field of prime characteristic and [E : K]> Ro. In ?2 we shall see that finitedimensional subspaces of straight locally compact vector spaces need not be discrete. In ?3 we shall see that the validity of the Open Mapping and Closed Graph theorems for straight locally compact vector spaces depends on whether those spaces are generated by compact subsets. In ?4 we shall study locally compact algebras of linear operators. In ?5 we shall obtain structure theorems for a commutative semisimple straight locally compact algebra over an infinite discrete field; such an algebra is the topological direct product of a discrete algebra and a local subdirect product of a sequence of algebras, each discrete and algebraically the cartesian product of a family of fields, relative to finite subfields.